Research Profile

My main focus is on K3 surfaces and higher dimensional analogues which can be studied in terms of algebraic invariants like Hodge structures and derived categories. K3 surfaces and related moduli spaces are particularly interesting test cases for some of the central conjectures in algebraic geometry (e.g. Tate, Hodge, Bloch-Beilinson). I have studied Chow groups of K3 surfaces from a geometric and a categorical perspective. In particular, I have introduced the notion of constant cycle curves and studied the action of symplectic automorphisms on Chow groups, providing further evidence for one of Bloch's elusive conjectures. Finite group of symplectic derived auto-equivalences have been classified completely in terms of the Conway group, one of the exotic sporadic simple groups. For Kuznetsov's K3 category associated with any cubic fourfold I have extended work of Addington and Thomas to the twisted case and described the group of auto-equivalences in the generic case. This has subsequently led to a new proof of the global Torelli theorem for cubic fourfolds (with Rennemo).

It has been conjecture that rationality of cubic fourfolds is determined by the structure of the associated K3 category. Further investigations of the structure of Kuznetsov's category should shed more light on the role of derived techniques on rationality questions in broader generality. The bearing of derived techniques on our understanding of cycles on K3 surfaces and cubics hypersurfaces needs to be clarified. Cohomological methods relating classical invariants like the Jacobian ring of a hypersurface with categorical invariants similar to Hochschild cohomology may lead to global Torelli theorems for cubics of higher dimensions. The role of mirror symmetry needs to be explored. Further foundational questions concerning the motivic nature of K3 surfaces shall be addressed.

Contribution to Research Areas

Research Area CHomological mirror symmetry relates symplectic and algebraic geometry as an equivalence of categories (Fukaya category of Lagrangians resp. derived category of coherent sheaves). Fundamental aspects of both sides can thus be seen also from the mirror perspective which has led to new insight. In [1], we have proved the mirror analogue of a theorem of Donaldson on the action of the diffeomorphism group of a K3 surface. The conjectured braid group like description of the group of autoequivalences of the derived category of Calabi-Yau varieties of dimension two is an example and one of the main open problems in the area.